Edge states and the bulk-boundary correspondence in Dirac Hamiltonians

We present an analytic prescription for computing the edge dispersion E(k) of a tight-binding Dirac Hamiltonian terminated at an abrupt crystalline edge. Specifically, we consider translationally invariant Dirac Hamiltonians with nearest-layer interaction. We present and prove a geometric formula that relates the existence of surface states as well as their energy dispersion to properties of the bulk Hamiltonian. We further prove the bulk-boundary correspondence between the Chern number and the chiral edge modes for quantum Hall systems within the class of Hamiltonians studied in the paper. Our results can be extended to the case of continuum theories which are quadratic in the momentum, as well as other symmetry classes.Comment: 8 pages + appendice

Topological Defects on the Lattice I: The Ising model

In this paper and its sequel, we construct topologically invariant defects in two-dimensional classical lattice models and quantum spin chains. We show how defect lines commute with the transfer matrix/Hamiltonian when they obey the defect commutation relations, cousins of the Yang-Baxter equation. These relations and their solutions can be extended to allow defect lines to branch and fuse, again with properties depending only on topology. In this part I, we focus on the simplest example, the Ising model. We define lattice spin-flip and duality defects and their branching, and prove they are topological. One useful consequence is a simple implementation of Kramers-Wannier duality on the torus and higher genus surfaces by using the fusion of duality defects. We use these topological defects to do simple calculations that yield exact properties of the conformal field theory describing the continuum limit. For example, the shift in momentum quantization with duality-twisted boundary conditions yields the conformal spin 1/16 of the chiral spin field. Even more strikingly, we derive the modular transformation matrices explicitly and exactly.Comment: 45 pages, 9 figure

Stability of zero modes in parafermion chains

One-dimensional topological phases can host localized zero-energy modes that enable high-fidelity storage and manipulation of quantum information. Majorana fermion chains support a classic example of such a phase, having zero modes that guarantee two-fold degeneracy in all eigenstates up to exponentially small finite-size corrections. Chains of `parafermions'---generalized Majorana fermions---also support localized zero modes, but, curiously, only under much more restricted circumstances. We shed light on the enigmatic zero mode stability in parafermion chains by analytically and numerically studying the spectrum and developing an intuitive physical picture in terms of domain-wall dynamics. Specifically, we show that even if the system resides in a gapped topological phase with an exponentially accurate ground-state degeneracy, higher-energy states can exhibit a splitting that scales as a power law with system size---categorically ruling out exact localized zero modes. The transition to power-law behavior is described by critical behavior appearing exclusively within excited states.Comment: 15 pages, 8 figures; substantial improvements to chiral case, coauthor added. Published 7 October 201

Replica topological order in quantum mixed states and quantum error correction

Topological phases of matter offer a promising platform for quantum computation and quantum error correction. Nevertheless, unlike its counterpart in pure states, descriptions of topological order in mixed states remain relatively under-explored. Our work give two definitions for replica topological order in mixed states, which involve $n$ copies of density matrices of the mixed state. Our framework categorizes topological orders in mixed states as either quantum, classical, or trivial, depending on the type of information that can be encoded. For the case of the toric code model in the presence of decoherence, we associate for each phase a quantum channel and describes the structure of the code space. We show that in the quantum-topological phase, there exists a postselection-based error correction protocol that recovers the quantum information, while in the classical-topological phase, the quantum information has decohere and cannot be fully recovered. We accomplish this by describing the mixed state as a projected entangled pairs state (PEPS) and identifying the symmetry-protected topological order of its boundary state to the bulk topology. We discuss the extent that our findings can be extrapolated to $n \to 1$ limit

Quantum dynamics of thermalizing systems

We introduce a method "DMT" for approximating density operators of 1D systems that, when combined with a standard framework for time evolution (TEBD), makes possible simulation of the dynamics of strongly thermalizing systems to arbitrary times. We demonstrate that the method performs well for both near-equilibrium initial states (Gibbs states with spatially varying temperatures) and far-from-equilibrium initial states, including quenches across phase transitions and pure states